Enhancement of Heat Transfer Using Pins Swimming in Non-Isothermal Fluidic Systems : Exact Solutions

Heat transfer to pins swimming in non-isothermal fluidic systems is theoretically analyzed. Four different cases are considered: [A] pins aligned longitudinally in flowing fluid having constant temperature gradient, [B] pins aligned transversely in flowing fluid flow with constant temperature gradient, [C] pins moving longitudinally towards a heated surface, and [D] pins moving transversely towards the heated surface. The Appropriate unsteady energy transport equations are solved and closed form solutions for the fin temperatures are obtained. Accordingly, different performance indicators are calculated. It is found that heat transfer to the swimming pin increases as the pin thermal length, Peclet number and fluid temperature difference along the pin length increase. It decreases as fluid temperature index along the motion direction increases. Moreover, the swimming pins of case C are found to produce the maximum system effective thermal conductivity. In addition, pins of case B with thermal lengths above 11 produce system thermal conductivity independent on the thermal length. Meanwhile, pins of case A having thermal lengths above 10 produce system thermal conductivities less responsive to the thermal length. The system thermal conductivity is noticed to increase as the thermal length and Peclet number increase. Eventually, pins of case D produce system thermal conductivities that are independent on the transverse temperature. Finally, the results of this work provide a basis for modeling super convective fluidic systems that can be used in cooling of electronic components.


Introduction
Enhancing heat transfer in thermal systems becomes the primary goal in thermal engineering.This goal can be achieved by various means such as using fins [1][2][3][4] and nanofluids [5].Researchers succeeded to enhance the heat transfer from the finned heated surfaces by several means.For example, many fins configurations such as the slit fins have produced heat transfer coefficient 50 to 100 percent above those of flat fins [6].Moreover, Yang et al. [7] succeeded to increase the heat transfer rate from a fin-tube heat exchanger by 6.3 percent by optimizing the fin spacing.Sahiti et al. [8] found out that employing interrupted elliptic fins could produce heat transfer coefficient 50 percent above that produced by interrupted circular fins.
Almogbel and Bejan [9] and Almogbel [10] proved that constructal optimization of a fin assembly is capable of producing heat transfer coefficient 500 percent above those of single fin under the same constraints.Li et al. [11] showed that microfinnned helical tubes caused an increase in the heat transfer by 120 percent compared to that of plain tube.Kiwan and Al-Nimr [12] were among the recent researchers who found that using a porous fin with certain porosity might give same performance as conventional fin and save 100 times porosity percent of the fin material.Aldoss et al. [13] found out that the heat transfer through a capsulated liquid metal fins might reach about 500% above that of an equal-size steel fin.Recently, Khaled [14,15] showed that heat transfer using joint and permeable fins could be 100% and 40% above that of typical flat fins, respectively.Moreover, Khaled [16,17] showed that fins with roots and fins with internal flows can increase heat transfer by about 100% and 200%, respectively, compared flat rootless fins.
Another recent method for enhancing the heat transfer from the heated surfaces is by using nanofluids.These are fluids containing suspensions of nanoparticles made of high thermally conductive materials.Nanofluids are noticed to possess large effective thermal conductivities for very low concentrations of nanoparticles.For example, the effective thermal conductivity of ethylene glycol is increased by about 40% above that of the base fluid when a 0.3% volume of copper nanoparticles of diameters less than 10 nm are suspended in the fluid [18].In addition, Xuan and Li [19] showed that the convective heat transfer coefficient was increased by 60% for nanofluid composed of water and 2% Cu-nanoparticles by volume.Finally, the estimated maximum levels of the heat transfer enhancement due to different recent enhancing methods are summarized in the work of Mansoor et al. [20].Among these methods are utilizing (a) fins and microfins, (b) porous media, (c) large particles suspensions, (d) nanofluids, (e) phase-change devices, (f) flexible seals, (g) flexible complex seals, (h) vortex generators, (i) protrusions, and (j) ultra high thermal conductivity composite materials.
From the previous brief literature survey, it is noticed that many novel ideas were developed recently to enhance the heat transfer from the solid surfaces.In addition, it is proposed in the present work to consider high thermally conductive pins submerged and swimming in the fluid as a new novel method for enhancing the heat transfer from the solid surface.The pin velocities are considered different from the fluid velocity.It is referred to these pins as to "swimming pins".The resulting solid-fluid system is different from the two-phase flow or the nanofluid system [20] by having the pins moving at constant or predetermined velocity.The velocity of the swimming pins can be attained using different methods.Among these methods are: 1) Applying appropriate magnetic forces on the swimming pins along with using magnetized pins, 2) Applying appropriate centrifugal forces on the fluid/suspended-pins system, and 3) Appling appropriate electrostatic forces on the swimming pins along with using electrically charged pins.To the best author knowledge, this proposal has not been discussed before and it is though to be useful for cooling of electronic components.
In this work, four different types of swimming pins are theoretically analyzed.These are the following: [A] Pins swimming longitudinally in flowing fluid having constant temperature gradient, [B] Pins aligned transversely and swimming in flowing fluid having constant temperature gradient, [C] Pins swimming longitudinally towards a heated surface, and [D] Pins moving transversely towards the heated surface.Such these cases can form a basis for constructing and modeling super convective fluid/swimming-pins systems.Appropriate unsteady en-ergy transport equations are identified and solved theoretically.Closed form solutions for the pin temperature are obtained for each case.Accordingly, different pin performance indicators and the effective thermal conductivity of the fluid/swimming-pins system are computed for the studied cases.

Pins Swimming Longitudinally in Flowing
Fluid Having Constant Temperature Gradient (Cases A) Consider a pin having a uniform cross-sectional area A C and a uniform perimeter P.This pin is suspended in flowing fluid inside a heated pipe of diameter 2r o .Moreover, the pin is considered to move inside the flowing fluid at a constant velocity U.The pin length L which is much larger than its characteristic transverse dimension L C = A C /P is taken for this case to be aligned along the x'-direction as shown in Figure 1.Both x' and U have the same direction.Let the pipe be subjected to constant heat flux s q and the mass flow rate of the flowing fluid be .As such, the surrounding fluid stream temperature d inside the pipe can be shown to be equal to the following under fully developed condition: where P o , c p , , d i , x' and x are the pipe perimeter T   o , fluid specific heat, surrounding fluid stream temperature at the inlet, distance between the inlet and the pin left end and the pin coordinate starting from its left end, respectively.The magnitude of x' is given by x' = Ut where t is the time variable.The one-dimensional energy equation applied to the pin can be written in the following form: where T f ,  f , c f , k f and h are the pin temperature, pin density, pin specific heat, pin thermal conductivity and the convection heat transfer between the pin and the surrounding fluid, respectively.Correlations for the h-coefficient for each case with sufficient accuracy can be obtained using any recent heat transfer textbooks [21].
Applying the following dimensionless variables: , on Equation (2) will change it to the following dimensionless form: where f Pe (pin Peclet number), m (pin index) and 1


(characteristic function) are given by ; ; The pin is considered to be insulated from both left and right ends.In addition, it is considered to be at temperature , f d i when t = 0.As such, the dimensionless boundaries and initial conditions are given by The net heat transfer rate from the pin to the flowing fluid, f d q  , is given by The temperature increase across any pipe differential element of thickness L which is denoted by T L is equal to  .Define the dimensionless pin heat transfer rate  A as the ratio between the pin heat transfer rate to that having maximum temperature difference of

Closed Form Solutions
Define the variable   * * , g x t as the following: Substituting Equation (9) in Equation ( 4) results in the following form of energy equation: The boundaries and initial conditions are obtained by substituting Equation (9)   The corresponding boundary and initial conditions are taken to be Utilizing the theory of ordinary differential equations, separation of variables and the Sturm-Liouville theory [17], the following solutions can be obtained: where The functions The constants E a and E n 's are equal to As such, the dimensionless heat transfer rate is equal to The maximum effective swimming pin index denoted by m  can be obtained by solving the following equation: 0.99 As such, m  L for this case is obtained by solving the following equation: where . Note that the maximum heat transfer rate is equal to

Pins Aligned Transversely and Swimming in Flowing Fluid Having Constant Temperature Gradient (Cases B)
Let the pin described in section 2.1 be aligned transversely such that the pin centerline is normal to the U-direction as shown in Figure 1.The fluid temperature around the pin can be approximated by the following expression: where d is the fluid undisturbed stream temperature increase across the pin length.Define the dimensionless time according to the following: According to Equation (23) and Equation (3a), Equation (22) The one-dimensional energy equation applied to the pin for this case has the following form: Utilizing the dimensionless variables given by Equation (3a,c) and Equation (23), Equation (25) can be changed to the following dimensionless form: where 2  (characteristic function) and 1  (Temperature difference ratio) are given by The dimensionless boundaries and initial conditions as adopted in section 2.1 can be expressed in the following dimensionless forms: The heat transfer from the pin to the flowing fluid is given by Define the dimensionless pin heat transfer rate  B as the ratio between the pin heat transfer rate to that having maximum temperature difference of .Mathematically, it is equal to

Closed Form Solutions
Redefine the variable   * * , g x t as the following: Substituting Equation (31) in Equation ( 25) results in the following dimensionless form of the energy equation: The boundaries and initial conditions are obtained by substituting Equation (31) in Equations (28a-c).They are given by Equations (11a-c).The equations for the steady state and homogenous solutions are given by   The corresponding boundary and initial conditions are given by Equations (14a-c).Utilizing the theory of ordinary differential equations, separation of variables and the Sturm-Liouville theory [22], the following solutions can be obtained.
As such, the dimensionless heat transfer rate is equal to The steady state value of B  is zero.According to Equation (35), the pin base section is located at where the convection heat transfer changes its direction at this section.Therefore, the maximum conduction heat transfer rate through this pin is obtained from the following mathematical relationship: As such, the pin maximum dimensionless conduction heat transfer denoted by  is equal to The effective swimming pin length denoted by L  can be obtained by solving the following equation: . As such, mL  can be shown to be .10.597 mL  

Pins Swimming Longitudinally Towards a Heated Surface (Cases C)
Let the pin described in section 2.1 be moving towards a hot surface with temperature T s at a constant velocity U as shown in Figure 2. The fluid stream temperature surrounding the pin can be approximated by the following expression: where T  , b, x' and x are the fluid far stream temperature, positive quantity, the position vector of the pin upper end and the pin axial coordinate starting from the pin upper end, respectively.The position vector x' is a time dependent quantity given by x' = x o -Ut where t is the time variable.In terms of dimensionless variables given by Equation (3a,b), Equation (41) changes to where B and C are equal to According to Equation (41), the dimensionless time  ,  , , Applying the dimensionless variables given by Equation (3a,b) and Equation (44) on Equation (2) will change it to the following dimensionless form: The dimensionless boundaries and initial conditions as adopted in section 2.1 can be expressed in the following dimensionless forms: 1, ; The heat transfer from the pin to the flowing fluid is given by Define the dimensionless pin heat transfer rate  C as the ratio between the pin heat transfer rate to that having maximum temperature difference of

Closed Form Solutions
The solution of  * * The corresponding boundary and initial conditions are taken to be Note that the maximum heat transfer rate can be obtained and it is equal to can be obtained by solving the following equation: 0.99

Pins Moving Transversely Towards the Heated Surface (Cases D)
Let the pin described in section 2.3 be aligned transversely and be moving at a constant velocity U towards the hot surface as shown in Figure 2. The heated surface temperature is considered to be varying exponentially with x s -axis that is coinciding with pin axial axis.This variation is according to the following relationship: where b 2 is a positive quantity.Accordingly, the fluid stream temperature surrounding the pin given by Equation ( 41) is equal to where T so is the temperature of the surface opposing the pin left end.In terms of dimensionless variables given by Equation (3a) and Equation (23), Equation (57) changes to where B 2 , C and  2 are equal to According to Equation (57), the dimensionless time required for the pin to reach the heated boundary is given by f .Applying the dimensionless variables given by Equation (3a), Equation (44) and Equation (23) on Equation (25) will change it to the following dimensionless form: The dimensionless boundaries and initial conditions as adopted in section 2.1 can be expressed in the following dimensionless forms: The heat transfer from the pin to the flowing fluid is given by Define the dimensionless pin heat transfer rate  D as the ratio between the pin heat transfer rate to that with maximum temperature difference of

Closed Form Solutions
According to Equation (62),  D is always increasing with time.As such, the maximum value of  D , denoted by , is computed at dimensionless time given by ; ; ; Equation ( 66) is reducible to the following expression: The maximum effective swimming pin index denoted by m  can be obtained by solving the following equation: 0.99 As such, the dimensionless effective length for this case m  L can be obtained to be equal to

Effective Thermal Conductivity for the Fluid Loaded With Swimming Pins of Kind (A)
Suppose that the pin volume fraction is equal to where n f is the number of swimming pins, V sf is the volume of a single pin and V is the fluid volume.
The maximum value of  is taken to be 1.5% which is expected to cause insignificant effect on the fluid flow pressure drop [5,19].The net heat flux received by the fluid can be shown to be equal to where A Co , k and r are the pipe cross-sectional area , fluid thermal conductivity and the pipe radial coordinate, respectively.For homogenous mixture of the swimming pins and the pure fluid, the mixture can be approximated as a continuum fluid with effective thermal conductivity k effA .For this case, the boundary heat flux, By solving Eqs. ( 68) and (69) simultaneously, k effA can be obtained and it is equal to where Pe A is the pure fluid Peclet number; 2 Note that u m is the pure fluid mean velocity.

Effective Thermal Conductivity for the Fluid Loaded With Swimming Pins of Kind (B)
The steady state rate of heat transfer through the pin is independent on the pin speed.As such, the effective thermal conductivity can be obtained by equating the transverse conduction across a given volume of fluid and swimming pins (V = A Co L) to that across a homogenous mixture with effective thermal conductivity k effB under the same temperature difference.Mathematically, this energy balance is given by   As such, k effB is given by: Based on effective pin length, Equation (72

Effective Thermal Conductivity for the Fluid Loaded With Swimming Pins of Kind (C)
The rate of heat transfer from the heated surface (per unit cross-sectional area normal to x'-axis) is given by: As such, k effC can be obtained and it is equal to

Effective Thermal Conductivity for the Fluid Loaded With Swimming Pins of Kind (D)
The rate of heat transfer from the heated surface (per unit cross-sectional area normal to x'-axis) is given by:   Therefore, k effD can be obtained and it is equal to Based on maximum effective pin index, Equation ( 76) is reducible to

Discussion of the Results
Figure 3 shows the variation of the dimensionless rate of heat transferred from the fluid to the pin A with the pin thermal length mL, pin Peclet number Pe f and dimensionless time t * for case A-swimming pins.Heat transfer to the swimming pin is expected to increase as mL increases however, the dominator of Equation ( 8) increases largely with the increase of hPL.The latter quaintly increases linearly with .As such,  is shown to decrease as mL increases.The initial value of A  is found from Figure 3 to be equal to 0.5.This value coincides with the value obtained by integrating Equation (11c) over the whole range of x * as given by Equation ( 8).This validates the obtained closed form solutions.As t * increases, A is found to increase at lower values of mL while at large values of mL, A   is unaffected with the time variable.Increasing the value of Pe f causes an increase in A  .This is because pins swimming at larger U-values (larger Pe f -values) reach to the fluid volumes with larger temperatures in a shorter period of time.Thus, the temperature difference between the fluid and the pin increases causing an increase in the rate of the heat transfer to the pin.
Factors that increases the convection heat transfer to the pin such as increasing the values of Pe f and mL and decreasing the time variable t * tend to increases the effective thermal conductivity, k effA , of the fluid/swimming-pins system.These trends are quite recognizable in Figure 4.It can be recommended from Figure 4 to have swimming pins with mL > 10 so that k effA be less sensitive to the variation of mL. Figure 5 shows the variation of the performance indicator  and effective thermal conductivity of the fluid/swimming-pins system, k effB , with the pin thermal length mL for case B-swimming pins.It is noticed that the values of  is almost independent on mL when mL < 0.65.In addition, it can be noticed that k effB increases as mL increases until it becomes independent on mL when mL > 11.Moreover, it can be seen that as the swimming pins volume ratio increases, the effective thermal conductivity k effB increases.
Figure 6 shows the effects of the pin thermal length mL, pin Peclet number Pe f and dimensionless fluid temperature variation index B on the maximum dimensionless rate of heat transferred from the fluid to the pin for case C-swimming pins.The role of mL and Pe f on max C is similar to their role on  which are discussed previously.For small B-values, the pin encounters a larger temperature difference between its temperature and the fluid temperature for a larger period of time as compared to cases with larger B-values.Thus, heat transfer to the swimming pin is expected to increase as B decreases.As such, max C increases as B decreases.The effective thermal conductivity k effC is found to be highly affected by Pe f and less affected by B at large mL-values as can be seen from Figure 7.According to this figure, the values of the effective thermal conductivity k effC can be larger than those for k effA and k effB at the same mL-values.Figure 8 shows that the case D-swimming pin maximum heat transfer is highly affected by the fluid temperature increase along the pin length.However, the effective thermal conductivity of the fluid/swimming-pins system, k effD , is noticed to be independent on that temperature increase as given by        A as B approaches zero.

Conclusions
In this work, heat transfer to pins swimming inside no-isothermal fluid was theoretically analyzed.Four dif-ferent types of swimming pins were considered: [A] Pins have thermal lengths larger than 10 so that the system effective thermal conductivity be less responsive to the thermal length. Swimming pins of case B with thermal lengths larger than 11 produce system effective thermal conductivity independent on the variation of the thermal length. Swimming pins of case C produces system effective thermal conductivities larger than those of cases A and B. This trend agrees with the finding of Khaled and Vafai [23] that boundary arrangement of the dispersive elements produce substantial enhancements in heat transfer compared to other arrangements.
 Swimming pins of case D produces system effective thermal conductivities independent on temperature difference along the pin length.Finally, the closed form solutions derived in this work can form a basis for constructing and modeling super convective fluidic systems that can be used in cooling of electronic components.

Figure 1 .
Figure 1.Schematic diagram for swimming pins of cases: A and B, and the flow configurations with coordinates system.

Figure 2 .
Figure 2. Schematic diagram for swimming pins of cases: C and D, and the flow configurations with coordinates system.required for the pin to reach the heated surface is equal to *

Figure 3 .
Figure 3. Effects of mL, Pe f and t * on Θ A .

Figure 4 .
Figure 4. Effects of mL, U/u m and t * on k effA /k.

Figure 5 .
Figure 5. Effects of mL on  an, mL and  on k effB /k.

Figure 6 .Figure 7 .
Figure 6.Effects of mL, Pe f and B on .  max   C

Figure 8 .Figure 9 .
Figure 8. Effects of mL, Pe f and B 2 on . max  D

Figure 10 .
Figure 10.Effects of Pe f and B on m  L. swimming longitudinally in flowing fluid having constant temperature gradient, [B] Pins aligned transversely and swimming in flowing fluid having constant temperature gradient, [C] Pins swimming longitudinally towards a heated surface, and [D] Pins moving transversely towards the heated surface.Appropriate unsteady energy transport equations were identified, dimesionlaized and solved theoretically.Closed form solutions for the pin temperature and the heat transfer rates to the pins or through the pins were obtained for each case.Accordingly, the effective thermal conductivities of the fluid/ swimming-pins systems were computed.The following main findings were reported:  Heat transfer to the swimming pin increases as the pin thermal length increases. Heat transfer to the swimming pin increases as the pin Peclet number increases. Heat transfer to the swimming pin increases as the fluid temperature index decreases. Heat transfer to the swimming pin increases as the temperature difference along the pin length increases. The fluid/swimming-pins system effective thermal conductivity increases as both pin thermal length and Peclet number increase. It can be recommended that swimming pins of case A have thermal lengths larger than 10 so that the system effective thermal conductivity be less responsive to the thermal length. Swimming pins of case B with thermal lengths larger than 11 produce system effective thermal conductivity independent on the variation of the thermal length. Swimming pins of case C produces system effective thermal conductivities larger than those of cases A and B. This trend agrees with the finding of Khaled and Vafai[23] that boundary arrangement of the dispersive elements produce substantial enhancements in heat transfer compared to other arrangements.